| 1 |
\chapter{\label{chap:ice}PHASE BEHAVIOR OF WATER IN COMPUTER \\ SIMULATIONS} |
| 2 |
|
| 3 |
As mentioned in the previous chapter, water has proven to be a |
| 4 |
challenging substance to depict in simulations, and a variety of |
| 5 |
models have been developed to describe its behavior under varying |
| 6 |
simulation |
| 7 |
conditions.\cite{Stillinger74,Rahman75,Berendsen81,Jorgensen83,Bratko85,Berendsen87,Caldwell95,Liu96,vanderSpoel98,Urbic00,Mahoney00,Fennell04} |
| 8 |
These models have been used to investigate important physical |
| 9 |
phenomena like phase transitions and the hydrophobic |
| 10 |
effect.\cite{Yamada02,Marrink94,Gallagher03} With the choice of models |
| 11 |
available, it is only natural to compare them under interesting |
| 12 |
thermodynamic conditions in an attempt to clarify the limitations of |
| 13 |
each.\cite{Jorgensen83,Jorgensen98b,Baez94,Mahoney01} Two important |
| 14 |
properties to quantify are the Gibbs and Helmholtz free energies, |
| 15 |
particularly for the solid forms of water, as these predict the |
| 16 |
thermodynamic stability of the various phases. Water has a |
| 17 |
particularly rich phase diagram and takes on a number of different |
| 18 |
stable crystalline structures as the temperature and pressure are |
| 19 |
varied. This complexity makes it a challenging task to investigate the |
| 20 |
entire free energy landscape.\cite{Sanz04} Ideally, research is |
| 21 |
focused on the phases having the lowest free energy at a given state |
| 22 |
point, because these phases will dictate the relevant transition |
| 23 |
temperatures and pressures for the model. |
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|
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The high-pressure phases of water (ice II-ice X as well as ice XII-ice XIV) |
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have been studied extensively both experimentally and |
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computationally. In this chapter, standard reference state methods |
| 28 |
were applied in the {\it low} pressure regime to evaluate the free |
| 29 |
energies for a few known crystalline water polymorphs that might be |
| 30 |
stable at these pressures. This work is unique in the fact that one of |
| 31 |
the crystal lattices was arrived at through crystallization of a |
| 32 |
computationally efficient water model under constant pressure and |
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temperature conditions. |
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|
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While performing a series of melting simulations on an early iteration |
| 36 |
of SSD/E, we observed several recrystallization events at a constant |
| 37 |
pressure of 1 atm. After melting from ice I$_\textrm{h}$ at 235~K, two |
| 38 |
of five systems recrystallized near 245~K. Crystallization events are |
| 39 |
interesting in and of themselves;\cite{Matsumoto02,Yamada02} however, |
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the crystal structure extracted from these systems is different from |
| 41 |
any previously observed ice polymorphs in experiment or |
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simulation.\cite{Fennell04} We have named this structure ``imaginary |
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ice'' (Ice-$i$) to indicate its origin in computer simulations. The |
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unit cell of Ice-$i$ and an axially elongated variant named |
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Ice-$i^\prime$ both consist of eight water molecules that stack in |
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rows of interlocking water tetramers as illustrated in figure |
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\ref{fig:iceiCell}. These tetramers form a crystal structure |
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similar in appearance to a recently simulated two-dimensional surface |
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tessellation of water on silica.\cite{Yang04} As expected in an ice |
| 50 |
crystal constructed of water tetramers, the hydrogen bonds are not as |
| 51 |
linear as those present in ice I$_\textrm{h}$; however, the |
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interlocking of these subunits appears to provide significant |
| 53 |
stabilization to the overall crystal. The arrangement of these |
| 54 |
tetramers results in open octagonal cavities that are typically |
| 55 |
greater than 6.3~\AA\ in diameter (see figure |
| 56 |
\ref{fig:protOrder}). This open structure leads to crystals that are |
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typically 0.07~g/cm$^3$ less dense than ice I$_\textrm{h}$. |
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|
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\begin{figure} |
| 60 |
\includegraphics[width=\linewidth]{./figures/unitCell.pdf} |
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\caption{Unit cells for (A) Ice-$i$ and (B) Ice-$i^\prime$, the |
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elongated variant of Ice-$i$. For Ice-$i$, the $a$ to $c$ |
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relation is given by $a = 2.1214c$, while for Ice-$i^\prime$, $a = |
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1.7850c$.} |
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\label{fig:iceiCell} |
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\end{figure} |
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|
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\begin{figure} |
| 69 |
\centering |
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\includegraphics[width=3.5in]{./figures/orderedIcei.pdf} |
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\caption{Image of a proton ordered crystal of Ice-$i$ looking |
| 72 |
down the (001) crystal face. The rows of water tetramers surrounded by |
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octagonal pores leads to a crystal structure that is significantly |
| 74 |
less dense than ice I$_\textrm{h}$.} |
| 75 |
\label{fig:protOrder} |
| 76 |
\end{figure} |
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|
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Results from our initial studies indicated that Ice-$i$ is the |
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minimum energy crystal structure for the single point water models |
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investigated (for discussions on these single point dipole models, see |
| 81 |
the previous chapter and related |
| 82 |
articles\cite{Fennell04,Liu96,Bratko85}). These earlier results only |
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considered energetic stabilization and neglected entropic |
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contributions to the overall free energy. To address this issue, we |
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have calculated the absolute free energy of this crystal using |
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thermodynamic integration and compared it to the free energies of ice |
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I$_\textrm{c}$ and ice I$_\textrm{h}$ (the common low-density ice |
| 88 |
polymorphs) and ice B (a higher density, but very stable crystal |
| 89 |
structure observed by B\'{a}ez and Clancy in free energy studies of |
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SPC/E).\cite{Baez95b} This work includes results for the water model |
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from which Ice-$i$ was crystallized (SSD/E) in addition to several |
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common water models (TIP3P, TIP4P, TIP5P, and SPC/E) and a reaction |
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field parametrized single point dipole water model (SSD/RF). The |
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axially elongated variant, Ice-$i^\prime$, was used in calculations |
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involving SPC/E, TIP4P, and TIP5P. The square tetramer in Ice-$i$ |
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distorts in Ice-$i^\prime$ to form a rhombus with alternating 85 and |
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95 degree angles. Under SPC/E, TIP4P, and TIP5P, this geometry is |
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better at forming favorable hydrogen bonds. The degree of rhomboid |
| 99 |
distortion depends on the water model used but is significant enough |
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to split the peak in the radial distribution function which |
| 101 |
corresponds to diagonal sites in the tetramers. |
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|
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\section{Methods and Thermodynamic Integration} |
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|
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Canonical ensemble ($NVT$) molecular dynamics calculations were |
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performed using the {\sc oopse} molecular mechanics |
| 107 |
package.\cite{Meineke05} The densities chosen for the simulations were |
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taken from isobaric-isothermal ($NPT$) simulations performed at 1~atm |
| 109 |
and 200~K. Each model (and each crystal structure) was allowed to |
| 110 |
relax for 300~ps in the $NPT$ ensemble before averaging the density to |
| 111 |
obtain the volumes for the $NVT$ simulations. All molecules were |
| 112 |
treated as rigid bodies, with orientational motion propagated using |
| 113 |
the symplectic {\sc dlm} integration method described in section |
| 114 |
\ref{sec:IntroIntegrate}. |
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|
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|
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We used thermodynamic integration to calculate the Helmholtz free |
| 118 |
energies ({\it A}) of the listed water models at various state |
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points. Thermodynamic integration is an established technique that has |
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been used extensively in the calculation of free energies for |
| 121 |
condensed phases of |
| 122 |
materials.\cite{Frenkel84,Hermens88,Meijer90,Baez95a,Vlot99}. This |
| 123 |
method uses a sequence of simulations during which the system of |
| 124 |
interest is converted into a reference system for which the free |
| 125 |
energy is known analytically ($A_0$). The difference in potential |
| 126 |
energy between the reference system and the system of interest |
| 127 |
($\Delta V$) is then integrated in order to determine the free energy |
| 128 |
difference between the two states: |
| 129 |
\begin{equation} |
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A = A_0 + \int_0^1 \left\langle \Delta V \right\rangle_\lambda d\lambda. |
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\end{equation} |
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Here, $\lambda$ is the parameter that governs the transformation |
| 133 |
between the reference system and the system of interest. For |
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crystalline phases, an harmonically-restrained (Einstein) crystal is |
| 135 |
chosen as the reference state, while for liquid phases, the ideal gas |
| 136 |
is taken as the reference state. Figure \ref{fig:integrationPath} |
| 137 |
shows an example integration path for converting a crystalline system |
| 138 |
to the Einstein crystal reference state. |
| 139 |
\begin{figure} |
| 140 |
\includegraphics[width=\linewidth]{./figures/integrationPath.pdf} |
| 141 |
\caption{An example integration path to convert an unrestrained |
| 142 |
crystal ($\lambda = 1$) to the Einstein crystal reference state |
| 143 |
($\lambda = 0$). Note the increase in samples at either end of the |
| 144 |
path to improve the smoothness of the curve. For reversible processes, |
| 145 |
conversion of the Einstein crystal back to the system of interest will |
| 146 |
give an identical plot, thereby integrating to the same result.} |
| 147 |
\label{fig:integrationPath} |
| 148 |
\end{figure} |
| 149 |
|
| 150 |
In an Einstein crystal, the molecules are restrained at their ideal |
| 151 |
lattice locations and orientations. Using harmonic restraints, as |
| 152 |
applied by B\'{a}ez and Clancy, the total potential for this reference |
| 153 |
crystal ($V_\mathrm{EC}$) is the sum of all the harmonic restraints, |
| 154 |
\begin{equation} |
| 155 |
V_\mathrm{EC} = \frac{K_\mathrm{v}r^2}{2} + \frac{K_\theta\theta^2}{2} + |
| 156 |
\frac{K_\omega\omega^2}{2}, |
| 157 |
\end{equation} |
| 158 |
where $K_\mathrm{v}$, $K_\mathrm{\theta}$, and $K_\mathrm{\omega}$ are |
| 159 |
the spring constants restraining translational motion and deflection |
| 160 |
of and rotation around the principle axis of the molecule respectively |
| 161 |
(see figure \ref{fig:waterSpring}). These spring constants are |
| 162 |
typically calculated from the mean-square displacements of water |
| 163 |
molecules in an unrestrained ice crystal at 200~K. For these studies, |
| 164 |
$K_\mathrm{v} = 4.29$~kcal~mol$^{-1}$~\AA$^{-2}$, $K_\theta\ = |
| 165 |
13.88$~kcal~mol$^{-1}$~rad$^{-2}$, and $K_\omega\ = |
| 166 |
17.75$~kcal~mol$^{-1}$~rad$^{-2}$. It is clear from figure |
| 167 |
\ref{fig:waterSpring} that the values of $\theta$ range from $0$ to |
| 168 |
$\pi$, while $\omega$ ranges from $-\pi$ to $\pi$. The partition |
| 169 |
function for a molecular crystal restrained in this fashion can be |
| 170 |
evaluated analytically, and the Helmholtz Free Energy ({\it A}) is |
| 171 |
given by |
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\begin{equation} |
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\begin{split} |
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A = E_m &- kT\ln\left(\frac{kT}{h\nu}\right)^3 \\ |
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&- kT\ln\left[\pi^\frac{1}{2}\left( |
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\frac{8\pi^2I_\mathrm{A}kT}{h^2}\right)^\frac{1}{2} |
| 177 |
\left(\frac{8\pi^2I_\mathrm{B}kT}{h^2}\right)^\frac{1}{2} |
| 178 |
\left(\frac{8\pi^2I_\mathrm{C}kT}{h^2}\right)^\frac{1}{2} |
| 179 |
\right] \\ |
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&- kT\ln\left[\frac{kT}{2(\pi K_\omega K_\theta)^{\frac{1}{2}}} |
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\exp\left(-\frac{kT}{2K_\theta}\right) |
| 182 |
\int_0^{\left(\frac{kT}{2K_\theta}\right)^\frac{1}{2}} |
| 183 |
\exp(t^2)\mathrm{d}t\right], |
| 184 |
\end{split} |
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\label{eq:ecFreeEnergy} |
| 186 |
\end{equation} |
| 187 |
where $2\pi\nu = (K_\mathrm{v}/m)^{1/2}$, and $E_m$ is the minimum |
| 188 |
potential energy of the ideal crystal.\cite{Baez95a} |
| 189 |
|
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The choice of an Einstein crystal reference state is somewhat |
| 191 |
arbitrary. Any ideal system for which the partition function is known |
| 192 |
exactly could be used as a reference point, as long as the system does |
| 193 |
not undergo a phase transition during the integration path between the |
| 194 |
real and ideal systems. Nada and van der Eerden have shown that the |
| 195 |
use of different force constants in the Einstein crystal does not |
| 196 |
affect the total free energy, and Gao {\it et al.} have shown that |
| 197 |
free energies computed with the Debye crystal reference state differ |
| 198 |
from the Einstein crystal by only a few tenths of a |
| 199 |
kJ~mol$^{-1}$.\cite{Nada03,Gao00} These free energy differences can |
| 200 |
lead to some uncertainty in the computed melting point of the solids. |
| 201 |
\begin{figure} |
| 202 |
\centering |
| 203 |
\includegraphics[width=3.5in]{./figures/rotSpring.pdf} |
| 204 |
\caption{Possible orientational motions for a restrained molecule. |
| 205 |
$\theta$ angles correspond to displacement from the body-frame {\it |
| 206 |
z}-axis, while $\omega$ angles correspond to rotation about the |
| 207 |
body-frame {\it z}-axis. $K_\theta$ and $K_\omega$ are spring |
| 208 |
constants for the harmonic springs restraining motion in the $\theta$ |
| 209 |
and $\omega$ directions.} |
| 210 |
\label{fig:waterSpring} |
| 211 |
\end{figure} |
| 212 |
|
| 213 |
In the case of molecular liquids, the ideal vapor is chosen as the |
| 214 |
target reference state. There are several examples of liquid state |
| 215 |
free energy calculations of water models present in the |
| 216 |
literature.\cite{Hermens88,Quintana92,Mezei92,Baez95b} These methods |
| 217 |
typically differ in regard to the path taken for switching off the |
| 218 |
interaction potential to convert the system to an ideal gas of water |
| 219 |
molecules. In this study, we applied one of the most convenient |
| 220 |
methods and integrated over the $\lambda^4$ path, where all |
| 221 |
interaction parameters are scaled equally by this transformation |
| 222 |
parameter. This method is reversible and provides results in excellent |
| 223 |
agreement with other established methods.\cite{Baez95b} |
| 224 |
|
| 225 |
The Helmholtz free energy error was determined in the same manner in |
| 226 |
both the solid and the liquid free energy calculations. At each point |
| 227 |
along the integration path, we calculated the standard deviation of |
| 228 |
the potential energy difference. Addition or subtraction of these |
| 229 |
deviations to each of their respective points and integrating the |
| 230 |
curve again provides the upper and lower bounds of the uncertainty in |
| 231 |
the Helmholtz free energy. |
| 232 |
|
| 233 |
Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and |
| 234 |
Lennard-Jones interactions were gradually reduced by a cubic switching |
| 235 |
function. By applying this function, these interactions are smoothly |
| 236 |
truncated, thereby avoiding the poor energy conservation which results |
| 237 |
from harsher truncation schemes. The effect of a long-range |
| 238 |
correction was also investigated on select model systems in a variety |
| 239 |
of manners. For the SSD/RF model, a reaction field with a fixed |
| 240 |
dielectric constant of 80 was applied in all |
| 241 |
simulations.\cite{Onsager36} For a series of the least computationally |
| 242 |
expensive models (SSD/E, SSD/RF, TIP3P, and SPC/E), simulations were |
| 243 |
performed with longer cutoffs of 10.5, 12, 13.5, and 15~\AA\ to |
| 244 |
compare with the 9~\AA\ cutoff results. Finally, the effects of using |
| 245 |
the Ewald summation were estimated for TIP3P and SPC/E by performing |
| 246 |
single configuration smooth particle-mesh Ewald ({\sc spme}) |
| 247 |
calculations for each of the ice polymorphs.\cite{Ponder87} The |
| 248 |
calculated energy difference in the presence and absence of {\sc spme} |
| 249 |
was applied to the previous results in order to predict changes to the |
| 250 |
free energy landscape. |
| 251 |
|
| 252 |
In addition to the above procedures, we also tested how the inclusion |
| 253 |
of the Lennard-Jones long-range correction affects the free energy |
| 254 |
results. The correction for the Lennard-Jones truncation was included |
| 255 |
by integration of the equation discussed in section |
| 256 |
\ref{sec:LJCorrections}. Rather than discuss its affect alongside the |
| 257 |
free energy results, we will just mention that while the correction |
| 258 |
does lower the free energy of the higher density states more than the |
| 259 |
lower density states, the effect is so small that it is entirely |
| 260 |
overwhelmed by the error in the free energy calculation. Since its |
| 261 |
inclusion does not influence the results, the Lennard-Jones correction |
| 262 |
was omitted from all the calculations below. |
| 263 |
|
| 264 |
\section{Initial Free Energy Results} |
| 265 |
|
| 266 |
\begin{table} |
| 267 |
\centering |
| 268 |
\caption{HELMHOLTZ FREE ENERGIES AND TRANSITION TEMPERATURES AT 1 |
| 269 |
ATMOSPHERE FOR SEVERAL WATER MODELS} |
| 270 |
|
| 271 |
\footnotesize |
| 272 |
\begin{tabular}{lccccccc} |
| 273 |
\toprule |
| 274 |
\toprule |
| 275 |
Water Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-$i$ & Ice-$i^\prime$ & $T_\textrm{m}$ (*$T_\textrm{s}$) & $T_\textrm{b}$\\ |
| 276 |
\cmidrule(lr){2-6} |
| 277 |
\cmidrule(l){7-8} |
| 278 |
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} & \multicolumn{2}{c}{(K)}\\ |
| 279 |
\midrule |
| 280 |
TIP5P & -11.85(3) & -11.86(2) & -11.96(2) & - & -12.29(2) & 266(7) & 337(4)\\ |
| 281 |
TIP4P & -11.84(3) & -12.04(2) & -12.08(3) & - & -12.33(3) & 262(6) & 354(4)\\ |
| 282 |
TIP3P & -11.41(2) & -11.23(3) & -11.82(3) & -12.30(3) & - & 269(7) & 357(4)\\ |
| 283 |
SPC/E & -12.87(2) & -13.05(2) & -13.26(3) & - & -13.55(2) & 299(6) & 396(4)\\ |
| 284 |
SSD/E & -11.27(2) & -11.19(4) & -12.09(2) & -12.54(2) & - & *355(4) & -\\ |
| 285 |
SSD/RF & -11.96(2) & -11.60(2) & -12.53(3) & -12.79(2) & - & 278(7) & 382(4)\\ |
| 286 |
\bottomrule |
| 287 |
\end{tabular} |
| 288 |
\label{tab:freeEnergy} |
| 289 |
\end{table} |
| 290 |
The calculated free energies of proton-ordered variants of three low |
| 291 |
density polymorphs (I$_\textrm{h}$, I$_\textrm{c}$, and Ice-$i$ or |
| 292 |
Ice-$i^\prime$) and the stable higher density ice B are listed in |
| 293 |
table \ref{tab:freeEnergy}. Ice B was included because it has been |
| 294 |
shown to be a minimum free energy structure for SPC/E at ambient |
| 295 |
conditions.\cite{Baez95b} In addition to the free energies, the |
| 296 |
relevant transition temperatures at standard pressure are also |
| 297 |
displayed in table \ref{tab:freeEnergy}. These free energy values |
| 298 |
indicate that Ice-$i$ is the thermodynamically preferred state for |
| 299 |
all of the investigated water models under the chosen simulation |
| 300 |
conditions. With the free energy at these state points, the |
| 301 |
Gibbs-Helmholtz equation was used to project to other state points and |
| 302 |
to build phase diagrams. Figures \ref{fig:tp3PhaseDia} and |
| 303 |
\ref{fig:ssdrfPhaseDia} are example phase diagrams built from the |
| 304 |
results for the TIP3P and SSD/RF water models. All other models have |
| 305 |
similar structure, although the crossing points between the phases |
| 306 |
move to different temperatures and pressures as indicated from the |
| 307 |
transition temperatures in table |
| 308 |
\ref{tab:freeEnergy}. It is interesting to note that ice |
| 309 |
I$_\textrm{h}$ (and ice I$_\textrm{c}$ for that matter) do not appear |
| 310 |
in any of the phase diagrams for any of the models. For purposes of |
| 311 |
this study, ice B is representative of the dense ice polymorphs. A |
| 312 |
recent study by Sanz {\it et al.} provides details on the phase |
| 313 |
diagrams for SPC/E and TIP4P at higher pressures than those studied |
| 314 |
here.\cite{Sanz04} |
| 315 |
\begin{figure} |
| 316 |
\centering |
| 317 |
\includegraphics[width=\linewidth]{./figures/tp3PhaseDia.pdf} |
| 318 |
\caption{Phase diagram for the TIP3P water model in the low pressure |
| 319 |
regime. The displayed $T_m$ and $T_b$ values are good predictions of |
| 320 |
the experimental values; however, the solid phases shown are not the |
| 321 |
experimentally observed forms. Both cubic and hexagonal ice I are |
| 322 |
higher in energy and don't appear in the phase diagram.} |
| 323 |
\label{fig:tp3PhaseDia} |
| 324 |
\end{figure} |
| 325 |
\begin{figure} |
| 326 |
\centering |
| 327 |
\includegraphics[width=\linewidth]{./figures/ssdrfPhaseDia.pdf} |
| 328 |
\caption{Phase diagram for the SSD/RF water model in the low pressure |
| 329 |
regime. Calculations producing these results were done under an |
| 330 |
applied reaction field. It is interesting to note that this |
| 331 |
computationally efficient model (over 3 times more efficient than |
| 332 |
TIP3P) exhibits phase behavior similar to the less computationally |
| 333 |
conservative charge based models.} |
| 334 |
\label{fig:ssdrfPhaseDia} |
| 335 |
\end{figure} |
| 336 |
|
| 337 |
We note that all of the crystals investigated in this study are ideal |
| 338 |
proton-ordered antiferroelectric structures. All of the structures |
| 339 |
obey the Bernal-Fowler rules and should be able to form stable |
| 340 |
proton-{\it disordered} crystals which have the traditional |
| 341 |
$k_\textrm{B}$ln(3/2) residual entropy at 0~K.\cite{Bernal33,Pauling35} |
| 342 |
Simulations of proton-disordered structures are relatively unstable |
| 343 |
with all but the most expensive water models.\cite{Nada03} Our |
| 344 |
simulations have therefore been performed with the ordered |
| 345 |
antiferroelectric structures which do not require the residual entropy |
| 346 |
term to be accounted for in the free energies. This may result in some |
| 347 |
discrepancies when comparing our melting temperatures to the melting |
| 348 |
temperatures that have been calculated via thermodynamic integrations |
| 349 |
of the disordered structures.\cite{Sanz04} |
| 350 |
|
| 351 |
Most of the water models have melting points that compare quite |
| 352 |
favorably with the experimental value of 273~K. The unfortunate |
| 353 |
aspect of this result is that this phase change occurs between |
| 354 |
Ice-$i$ and the liquid state rather than ice I$_\textrm{h}$ and the liquid |
| 355 |
state. These results do not contradict other studies. Studies of ice |
| 356 |
I$_\textrm{h}$ using TIP4P predict a $T_m$ ranging from 191 to 238~K |
| 357 |
(differences being attributed to choice of interaction truncation and |
| 358 |
different ordered and disordered molecular |
| 359 |
arrangements).\cite{Nada03,Vlot99,Gao00,Sanz04} If the presence of ice |
| 360 |
B and Ice-$i$ were omitted, a $T_\textrm{m}$ value around 200~K |
| 361 |
would be predicted from this work. However, the $T_\textrm{m}$ from |
| 362 |
Ice-$i$ is calculated to be 262~K, indicating that these |
| 363 |
simulation-based structures ought to be included in studies probing |
| 364 |
phase transitions with this model. Also of interest in these results |
| 365 |
is that SSD/E does not exhibit a melting point at 1 atm but does |
| 366 |
sublime at 355~K. This is due to the significant stability of |
| 367 |
Ice-$i$ over all other polymorphs for this particular model under |
| 368 |
these conditions. While troubling, this behavior resulted in the |
| 369 |
spontaneous crystallization of Ice-$i$ which led us to investigate |
| 370 |
this structure. These observations provide a warning that simulations |
| 371 |
of SSD/E as a ``liquid'' near 300~K are actually metastable and run |
| 372 |
the risk of spontaneous crystallization. However, when a longer |
| 373 |
cutoff radius is used, SSD/E prefers the liquid state under standard |
| 374 |
temperature and pressure. |
| 375 |
|
| 376 |
\section{Effects of Potential Truncation} |
| 377 |
|
| 378 |
\begin{figure} |
| 379 |
\centering |
| 380 |
\includegraphics[width=\linewidth]{./figures/cutoffChange.pdf} |
| 381 |
\caption{Free energy as a function of cutoff radius for SSD/E, TIP3P, |
| 382 |
SPC/E, SSD/RF with a reaction field, and the TIP3P and SPC/E models |
| 383 |
with an added Ewald correction term. Error for the larger cutoff |
| 384 |
points is equivalent to that observed at 9.0~\AA\ (see Table |
| 385 |
\ref{tab:freeEnergy}). Data for ice I$_\textrm{c}$ with TIP3P using |
| 386 |
both 12 and 13.5~\AA\ cutoffs were omitted because the crystal was |
| 387 |
prone to distortion and melting at 200~K. Ice-$i^\prime$ is the |
| 388 |
form of Ice-$i$ used in the SPC/E simulations.} |
| 389 |
\label{fig:incCutoff} |
| 390 |
\end{figure} |
| 391 |
For the more computationally efficient water models, we have also |
| 392 |
investigated the effect of potential truncation on the computed free |
| 393 |
energies as a function of the cutoff radius. As seen in figure |
| 394 |
\ref{fig:incCutoff}, the free energies of the ice polymorphs with |
| 395 |
water models lacking a long-range correction show significant cutoff |
| 396 |
radius dependence. In general, there is a narrowing of the free |
| 397 |
energy differences while moving to greater cutoff radii. As the free |
| 398 |
energies for the polymorphs converge, the stability advantage that |
| 399 |
Ice-$i$ exhibits is reduced. Adjacent to each of these plots are |
| 400 |
results for systems with applied or estimated long-range corrections. |
| 401 |
SSD/RF was parametrized for use with a reaction field, and the benefit |
| 402 |
provided by this computationally inexpensive correction is apparent. |
| 403 |
The free energies are largely independent of the size of the reaction |
| 404 |
field cavity in this model, so small cutoff radii mimic bulk |
| 405 |
calculations quite well under SSD/RF. |
| 406 |
|
| 407 |
Although the point-charge models investigated here were parametrized |
| 408 |
for use without the Ewald summation, we have estimated the effect of |
| 409 |
this method for computing long-range electrostatics for both TIP3P and |
| 410 |
SPC/E. This was accomplished by calculating the potential energy of |
| 411 |
identical crystals both with and without {\sc spme}. Similar behavior |
| 412 |
to that observed with reaction field is seen for both of these models. |
| 413 |
The free energies show reduced dependence on cutoff radius and span a |
| 414 |
narrower range for the various polymorphs. Like the dipolar water |
| 415 |
models, TIP3P displays a relatively constant preference for the |
| 416 |
Ice-$i$ polymorph. The crystal preference is much more difficult to |
| 417 |
determine for SPC/E. Without a long-range correction, each of the |
| 418 |
polymorphs studied assumes the role of the thermodynamically preferred |
| 419 |
state under different cutoff radii. The inclusion of the Ewald |
| 420 |
correction flattens and narrows the gap in free energies such that the |
| 421 |
polymorphs are isoenergetic within statistical uncertainty. This |
| 422 |
suggests that other conditions, such as the density in fixed-volume |
| 423 |
simulations, can influence the polymorph expressed upon |
| 424 |
crystallization. |
| 425 |
|
| 426 |
\section{Expanded Results Using Damped Shifted Force Electrostatics} |
| 427 |
|
| 428 |
In chapter \ref{chap:electrostatics}, we discussed in detail a |
| 429 |
pairwise method for handling electrostatics (shifted force, {\sc sf}) |
| 430 |
that can be used as a simple and efficient replacement for the Ewald |
| 431 |
summation. Answering the question of the free energies of these ice |
| 432 |
polymorphs with varying water models would be an interesting |
| 433 |
application of this technique. To this end, we set up thermodynamic |
| 434 |
integrations of all of the previously discussed ice polymorphs using |
| 435 |
the {\sc sf} technique with a cutoff radius of 12~\AA\ and an $\alpha$ |
| 436 |
of 0.2125~\AA . These calculations were performed on TIP5P-E and |
| 437 |
TIP4P-Ew (variants of the root models optimized for the Ewald |
| 438 |
summation) as well as SPC/E, SSD/RF, and TRED (see section |
| 439 |
\ref{sec:tredWater}). |
| 440 |
|
| 441 |
\begin{table} |
| 442 |
\centering |
| 443 |
\caption{HELMHOLTZ FREE ENERGIES OF ICE POLYMORPHS USING THE DAMPED |
| 444 |
SHIFTED FORCE CORRECTION} |
| 445 |
\begin{tabular}{ lccccc } |
| 446 |
\toprule |
| 447 |
\toprule |
| 448 |
Model & I$_\textrm{h}$ & I$_\textrm{c}$ & B & Ice-$i$ & Ice-$i^\prime$ \\ |
| 449 |
\cmidrule(lr){2-6} |
| 450 |
& \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\ |
| 451 |
\midrule |
| 452 |
TIP5P-E & -11.98(4) & -11.96(4) & -11.87(3) & - & -11.95(3) \\ |
| 453 |
TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\ |
| 454 |
SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\ |
| 455 |
SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\ |
| 456 |
TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\ |
| 457 |
\bottomrule |
| 458 |
\end{tabular} |
| 459 |
\label{tab:dampedFreeEnergy} |
| 460 |
\end{table} |
| 461 |
The results of these calculations in table \ref{tab:dampedFreeEnergy} |
| 462 |
show similar behavior to the Ewald results in figure |
| 463 |
\ref{fig:incCutoff}, at least for SSD/RF and SPC/E which are present |
| 464 |
in both. The Helmholtz free energies of the ice polymorphs for SSD/RF |
| 465 |
order in the same fashion; however, Ice-$i$ and ice B are quite a bit |
| 466 |
closer in free energy (nearly isoenergetic). The free energy |
| 467 |
differences between ice polymorphs for TRED water parallel SSD/RF, |
| 468 |
with the exception that ice B is destabilized such that the free |
| 469 |
energy is not very close to Ice-$i$. The SPC/E results really show the |
| 470 |
near isoenergetic behavior when using the electrostatic |
| 471 |
correction. Ice B has the lowest Helmholtz free energy; however, all |
| 472 |
the polymorph results overlap within error. |
| 473 |
|
| 474 |
The most interesting results from these calculations come from the |
| 475 |
more expensive TIP4P-Ew and TIP5P-E results. Both of these models were |
| 476 |
optimized for use with an electrostatic correction and are |
| 477 |
geometrically arranged to mimic water following two different |
| 478 |
ideas. In TIP5P-E, the primary location for the negative charge in the |
| 479 |
molecule is assigned to the lone-pairs of the oxygen, while TIP4P-Ew |
| 480 |
places the negative charge near the center-of-mass along the H-O-H |
| 481 |
bisector. There is some debate as to which is the proper choice for |
| 482 |
the negative charge location, and this has in part led to a six-site |
| 483 |
water model that balances both of these options.\cite{Vega05,Nada03} |
| 484 |
The limited results in table \ref{tab:dampedFreeEnergy} support the |
| 485 |
results of Vega {\it et al.}, which indicate the TIP4P charge location |
| 486 |
geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew |
| 487 |
water model, the experimentally observed polymorph (ice |
| 488 |
I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly |
| 489 |
higher in energy, though overlapping within error. Additionally, the |
| 490 |
less realistic ice B and Ice-$i^\prime$ structures are destabilized |
| 491 |
relative to these polymorphs. TIP5P-E shows similar behavior to SPC/E, |
| 492 |
where there is no real free energy distinction between the various |
| 493 |
polymorphs, because many overlap within error. While ice B is close in |
| 494 |
free energy to the other polymorphs, these results fail to support the |
| 495 |
findings of other researchers indicating the preferred form of TIP5P |
| 496 |
at 1~atm is a structure similar to ice |
| 497 |
B.\cite{Yamada02,Vega05,Abascal05} It should be noted that we are |
| 498 |
looking at TIP5P-E rather than TIP5P, and the differences in the |
| 499 |
Lennard-Jones parameters could be a reason for this dissimilarity. |
| 500 |
Overall, these results indicate that TIP4P-Ew is a better mimic of |
| 501 |
real water than these other models when studying crystallization and |
| 502 |
solid forms of water. |
| 503 |
|
| 504 |
\section{Conclusions} |
| 505 |
|
| 506 |
In this work, thermodynamic integration was used to determine the |
| 507 |
absolute free energies of several ice polymorphs. The new polymorph, |
| 508 |
Ice-$i$ was observed to be the stable crystalline state for {\it all} |
| 509 |
the water models when using a 9.0~\AA\ cutoff. Additional work showed |
| 510 |
that the free energy depends in part on simulation conditions - in |
| 511 |
particular, the choice of long-range correction method. Regardless, |
| 512 |
Ice-$i$ was still observed to be a stable polymorph for all of the |
| 513 |
studied water models. |
| 514 |
|
| 515 |
So what is the preferred solid polymorph for simulated water? The |
| 516 |
answer appears to be dependent both on the conditions and the model |
| 517 |
used. In the case of short cutoffs without a long-range interaction |
| 518 |
correction, Ice-$i$ and Ice-$i^\prime$ have the lowest free energy of |
| 519 |
the studied polymorphs with all the models. Ideally, crystallization |
| 520 |
of each model under constant pressure conditions, as was done with |
| 521 |
SSD/E, would aid in identifying their respective preferred structures. |
| 522 |
This work, however, helps illustrate how studies involving one |
| 523 |
specific model can lead to insight about important behavior of others. |
| 524 |
|
| 525 |
We also note that none of the water models used in this study are |
| 526 |
polarizable or flexible models. It is entirely possible that the |
| 527 |
polarizability of real water makes the Ice-$i$ structure substantially |
| 528 |
less stable than ice I$_\textrm{h}$. The dipole moment of the water |
| 529 |
molecules increases as the system becomes more condensed, and the |
| 530 |
increasing dipole moment should destabilize the tetramer structures in |
| 531 |
Ice-$i$. Right now, using TIP4P-Ew with an electrostatic correction |
| 532 |
gives the proper thermodynamically preferred state, and we recommend |
| 533 |
this arrangement for study of crystallization processes if the |
| 534 |
computational cost increase that comes with including polarizability |
| 535 |
is an issue. |
| 536 |
|
| 537 |
Finally, the stability of Ice-$i$ in these simulations raises the |
| 538 |
possibility of experimental observation. The extensive body of |
| 539 |
research on water in the low pressure regime makes us hesitant to |
| 540 |
ascribe any relevance to this work outside the simulation community. |
| 541 |
It is for this reason that we chose a name for this polymorph |
| 542 |
involving an imaginary quantity. That said, there are certain |
| 543 |
conditions that would be ideal for experimental observation of |
| 544 |
Ice-$i$. These include the negative pressure or stretched solid |
| 545 |
regime, clusters deposited in vacuum environments, and clathrate |
| 546 |
structures involving small non-polar molecules. For the purpose of |
| 547 |
comparison with future experimental results, we calculated the |
| 548 |
oxygen-oxygen pair correlation function, $g_\textrm{OO}(r)$, and the |
| 549 |
structure factor, $S(\vec{q})$ for the two Ice-$i$ variants (along |
| 550 |
with ice I$_\textrm{h}$ and I$_\textrm{c}$) at 77~K (figures |
| 551 |
\ref{fig:gofr} and \ref{fig:sofq}). It is interesting to note that |
| 552 |
the structure factors for Ice-$i^\prime$ and ice I$_\textrm{c}$ are |
| 553 |
quite similar. The primary differences are small peaks at 1.125, |
| 554 |
2.29, and 2.53~\AA$^{-1}$, so particular attention to these regions |
| 555 |
would be needed to distinguish Ice-$i^\prime$ from ice I$_\textrm{c}$. |
| 556 |
|
| 557 |
\begin{figure} |
| 558 |
\includegraphics[width=\linewidth]{./figures/iceGofr.pdf} |
| 559 |
\caption{Radial distribution functions of Ice-$i$ and ice |
| 560 |
I$_\textrm{c}$ calculated from from simulations of the SSD/RF water |
| 561 |
model at 77~K.} |
| 562 |
\label{fig:gofr} |
| 563 |
\end{figure} |
| 564 |
|
| 565 |
\begin{figure} |
| 566 |
\includegraphics[width=\linewidth]{./figures/sofq.pdf} |
| 567 |
\caption{Predicted structure factors for Ice-$i$ and ice |
| 568 |
I$_\textrm{c}$ at 77~K. The raw structure factors have been |
| 569 |
convoluted with a Gaussian instrument function (0.075~\AA$^{-1}$ |
| 570 |
width) to compensate for the truncation effects in our finite size |
| 571 |
simulations.} |
| 572 |
\label{fig:sofq} |
| 573 |
\end{figure} |
| 574 |
|
